solubility of nitrogen in ferrite; the fe–n phase diagram

Jendrik Steina , Ralf Erich Schacherla , Minsu Jungb , Sairamudu Mekab , Bastian Rheingansa ,Eric Jan Mittemeijera,ba Institute for Materials Science, University of Stuttgart, Stuttgart, Germanyb Max Planck Institute for Intelligent Systems (formerly Max Planck Institute for Metals Research), Stuttgart, Germany

Solubility of nitrogen in ferrite; the Fe–N phasediagram

To accurately define important phase boundaries in theiron–nitrogen (temperature–composition) phase diagramas well as the (temperature–potential) Lehrer diagram, thesolubility of nitrogen in ferrite was determined as a functionof the nitriding potential (which defines the chemical poten-tial of nitrogen) and the temperature. To this end, thin ironfoils were homogeneously nitrided in flowing gas mixturescomposed of ammonia and hydrogen. Phase identificationwas performed by means of X-ray diffraction analysis.Further, from the data obtained, the absorption functionand the enthalpy for dissolution of nitrogen into ferrite andthe enthalpy of the reaction occurring at the a/(a + c’)-phase boundary were determined. The data obtained werecorrected for the occurrence of a stationary state instead ofa local equilibrium at the surface of the specimens. It fol-lowed that parts of the phase boundaries in the Lehrer dia-gram do not represent equilibrium states but rather station-ary states.

Keywords: Nitrogen solubility; Lehrer diagram; Fe–Nphase diagram; Local equilibrium; Stationary state

1. Introduction

Next to the Fe–C phase diagram, the Fe–N phase diagramis one of the most important phase diagrams in materialsscience and engineering [1]. Thus a technologically veryimportant process of surface engineering of iron-based al-loys is based on \nitriding", i. e. the introduction of nitro-

gen into the specimen/component from an outer atmo-sphere (gas, plasma, salt) [2].From a scientific point of view, precise knowledge of the

thermodynamic properties of elements dissolved intersti-tially in a parent lattice is required, e. g. in view of the dif-ferent (conflicting) discussions of the thermodynamics ofsuch phases (see below for the Fe–N system).Phase-stability regions for the Fe–N system are usually

presented in a normal phase diagram (temperature vs. nitro-gen content) [3] and in particular also in a so-called Lehrerdiagram (temperature vs. nitriding potential, a direct mea-sure for the chemical potential of nitrogen; see section 2)[4]. As follows from the data used in [3] and from the yearof publication of the Lehrer diagram, the experimentswhich yielded these data were partly performed decadesago and present some ambiguity regarding certain impor-tant parts of these diagrams. This holds for the temperatureand composition at the end-point of the eutectoid line (a-Fe end of eutectoid line), where the a/(a + c) and a/(a + c’) phase1 boundaries meet in the Fe–N phase dia-gram, as well as for the temperature and nitriding potentialof the triple point, where the a, c and c’ phases meet in theLehrer diagram.Against the above background, the aim of the present

work is to provide accurate data for the equilibrium of

J. Stein et al.: Solubility of nitrogen in ferrite; the Fe–N phase diagram

Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 11 1053

International Journal ofMATERIALS RESEARCH

Zeitschrift fur METALLKUNDE OOriginal Contributions

1 a-Fe[N] = nitrogen ferrite (based on a bcc Fe-lattice with N (disor-dered) at octahedral interstices); c-Fe[N] = nitrogen austenite (basedon an fcc Fe-lattice with N (disordered) at octahedral interstices);c’ = Fe4N1–x (based on an fcc Fe-sublattice with N (ordered) at octahe-dral interstices); e = Fe2N1–x (based on an hcp Fe-sublattice with N(more or less ordered) at octahedral interstices).

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Fe–N phases in contact with a nitriding medium of fixedand known chemical potential of nitrogen2; i. e. precise de-termination of the above mentioned regions in the Fe–Nphase diagram and the Lehrer diagram. As a further result(i) the nitrogen absorption function for a-Fe[N] is obtainedand compared with descriptions given in the literature, and(ii) values are obtained for the enthalpy of absorption of ni-trogen in ferrite and the enthalpy of the reaction occurringat the a/(a + c’) phase boundary.In Fe–N phase-stability studies performed until now it

has not been sufficiently well appreciated that apparent\equilibrium states" can in fact be \stationary states" moreor less deviating from genuine equilibria. This paper in par-ticular also addresses this problem (see Section 3).

2. Thermodynamics

The nitriding of a metallic solid, M, in an NH3–H2 gas mix-ture can formally be imagined as the result of bringing N2gas into contact with M under a certain pressure. This state-ment is a consequence of the Gibbs energy (and thus thechemical potential) being a state variable. Therefore, theroute followed to reach a certain (final) state is irrelevantfor the value of the Gibbs energy of that (final) state. Thus,nitriding in an NH3–H2 gas mixture can be conceived asthe sum of the following (hypothetical) reactions:

1=2N2 ! ½N ð1aÞNH3 ! 1=2N2 þ 3=2H2 ð1bÞ_______________________+givingNH3 ! ½N þ 3=2H2 ð2Þ

where [N] represents N dissolved in the solid substrate M.Establishment of the equilibrium (2) implies the occurrenceof \local equilibrium" at (only) the surface of the substrateM (see footnote in Section 1).Equation (1a) implies that

1=2lN2;g ¼ lN;s ð3Þwhere lN2;g and lN;s represent the chemical potentials fornitrogen in the gas phase and the solid phase, respectively.Assuming ideal gases, or, at least, adopting a constant fuga-city coefficient3, it follows from Eq. (3):

1=2l0N2;g þ 1=2RT lnpN2p0

¼ l0N;s þ RT ln aN;s ð4Þ

where l0i (i = N2,g or Ns) is the chemical potential of compo-nent i in the reference state denoted by the superscript \0"(temperature dependent at the selected pressure of the refer-

ence state, p0, taken equal for all gas components, see be-low), pN2 is the partial pressure of the (hypothetical) N2gas in Eqs. (1a) and (1b) and aN,s is the activity of N in solidM. Now l0N;s is selected such that l

0N;s ¼ 1=2l0N2;g. Then it is

obtained:

aN;s ¼ K 1að Þ pN21=2.p0

1=2 ð5aÞ

since K(1a) = 1 due to the selected reference state. Using theequilibrium constants of Eqs. (1b) and (2),

Kð1bÞ ¼ pN21=2pH23=2.pNH3

p01and

Kð2Þ ¼ aN;s pH23=2.pNH3

p01=2

;

it follows

aN;s ¼ K 1bð Þ p01=2

rN ¼ K 2ð Þ p01=2

rN ð5bÞ

with the so-called nitriding potential rN given by

rN pNH3

.p3=2H2 ð6Þ

where pNH3 and pH2 are the partial pressures of the gas com-ponents NH3 and H2.The pressure of the reference state is selected as one pres-

sure unit (usually one atm), requiring that the partial pres-sures of all gas components in the equations have to be ex-pressed in the same unit as the pressure of the referencestate. As a consequence of this step the numerical value ofthe nitrogen activity in M can be in interpreted as the squareroot of the pressure (in pressure units, usually atm) of thehypothetical nitrogen gas occurring in Eqs. (1a) and (1b).The dependence of nitrogen activity on composition is dif-ferent for each Fe–N phase. Such relations can be deter-mined experimentally from so-called absorption isothermswhich express the relation between nitrogen content and ni-triding potential at constant temperature.It follows from the above discussion that phase fields

(phase-stability regions) for the Fe–N system can be pre-sented not only in the usual Fe–N phase diagram (T–cNdiagram) but also in a T– rN diagram: the so-called (poten-tial) Lehrer diagram.Because the solubility of N in a-Fe is very small (maxi-

mally about 0.4 at.%; see the result presented in this paper(Section 4.3)), Henry’s law can be adopted [5] for the nitro-gen activity. Then it can be written that

cN;a ¼ krN ð7Þ

where cN,a in the nitrogen concentration in a-Fe and k isgiven by the quotient of K(2) · (p

0)1/2 and the constant activ-ity coefficient.Thus, for the equilibrium between a-Fe and another Fe–

N phase i (i, for example, represents c’ or c) and the asso-ciated NH3–H2 gas mixture it holds:

cN;a ¼ krN;a=i ð8Þ

where rNa/i is the nitriding potential at a specific tempera-ture corresponding with equilibrium of a and i.According to chemical thermodynamics the equilibrium

constant K of a certain reaction obeys the expression


1054 Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 11

2 At the applied nitriding temperatures and pressure (normally 1 atm)the iron–nitrogen phases, such as a, c, c’ and e, are non-equilibriumphases: they are prone to decomposition into iron and nitrogen gas.Equilibrium at such (temperature and pressure) conditions can only berealized at the surface of the specimens by contact with a nitridingmedium of fixed chemical potential of nitrogen. Thus the usual Fe–Nphase diagram and the Lehrer diagram (ideally; cf. Section 3) repre-sent the equilibrium between Fe and a medium of largely variablechemical potential of nitrogen (for detailed discussion see [5] and[6]). Similar statements can be made for the Fe–C diagram: the solidcarbon phase in equilibrium with carbon ferrite (= a-Fe[C]) is graphiteand not cementite.3 In the latter case the fugacity coefficient is thought to be incorpo-rated in the reference chemical potential l0N2;g.

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RT lnK = –DG0 [7] with DG0 as the difference of the Gibbsenergies of pure products and pure reactants, according tothe reaction considered, in the reference states. BecauseDG0 = DH0 – TDS0, with DH0 and DS0 as the correspondingenthalpy and entropy (differences) of the reaction consid-ered, it holds that d(lnK)/d(1/T) = –DH0/R for many reac-tions, where DH0 (and also DS0) is practically constant overa large temperature range. As cN,a is proportional to K(2) (cf.Eq. (7)) it follows from the above that

d ln cN;a rNd 1=Tð Þ ¼ DH0N!a

Rð9Þ

with DH0N!a as the so-called enthalpy of dissolution of N ina (cf. Eq. (2)). Eq. (9) will be called the absorption functionin the following.The equilibrium of a and c’ (and of the associated NH3–

H2 gas mixture, not explicitly exhibited by Eq. (10)) can bewritten as

4Feþ ð1 xÞ½N aFe4N1 x ð10Þ

Recognizing that x 1, the corresponding equilibriumconstant can be written as

Kð10Þ ¼1

aN;a=c0ð11Þ

where aN,a/c’ denotes the activity of nitrogen in a in equilib-rium with c’. Again applying Henry’s law for N in a-Fe (seeabove), it is obtained

cN;a=c0 ¼ k0 ð12Þwhere k’ is given by the reciprocal of the product of K(10)and the constant activity coefficient. A similar treatment asabove for Eq. (9) then yields

d lnðcN;a=c0 Þdð1=TÞ ¼ DH0a=c0

Rð13Þ

with DHa=c0 as the enthalpy of reaction (10).

3. Local equilibrium versus stationary state

To obtain nitrogen-content data in the solid specimen repre-senting equilibrium with the nitriding medium for a seriesof values of the chemical potential of nitrogen, the chemicalpotential of nitrogen in the nitriding atmosphere has to becontrolled, i. e. it should be set and maintained at a certain,selected value. In the context of the treatment in Section 2,it then follows that control of the nitriding potential in anNH3–H2 gas mixture is imperative. This can be achievedby nitriding in a flow of an NH3–H2 gas mixture that is ofa fixed composition (corresponding to the desired nitridingpotential). If in the gas atmosphere equilibrium at 1 atmand at temperatures above 600 8C occurs, ammonia will bepractically fully dissociated. Hence, a stationary gas atmo-sphere is inappropriate. As this dissociation is a relativelyslow process, but catalytically activated by the presence ofiron (the specimen and, possibly, the furnace walls; seeFig. 4 in Ref. [6]), this thermal dissociation in the case ofnitriding iron-based specimens can be made negligible byapplication of a sufficiently large (linear) gas flowrate inthe furnace (see Section 4). However, if the nitriding poten-

tial has to be very large (i. e. for an NH3–H2 gas mixtureimplying a fraction of NH3 approaching 100%), even a tinyamount of thermal dissociation of ammonia causes the realnitriding potential to deviate distinctly from the one calcu-lated from the gas composition at the furnace inlet. Such asituation is not expected for the nitriding of iron-based ma-terials, but for example happens upon nitriding nickel in or-der to produce a Ni3N compound layer requiring an extre-mely large nitriding potential (e. g. performing nitridingwith a gas composed of pure ammonia at the gas inlet, whichcorresponds with rN =? at the gas inlet (cf. Eq. (6)) [8].Further kinetic conditions to be obeyed in order that the

above approach can be applied are:(i) equilibrium (2) is established;(ii) other gas components potentially present in the fur-

nace should behave as inert gases. This for exampleholds for the presence of N2, which may have beenadded to the gas atmosphere: it can be taken for certainthat equilibrium (1a) is not established;

(iii) the solid Fe–N phase(s) in the solid, which is (are) notlocated at the very surface, is (are) not in thermody-namic equilibrium and therefore this (these) Fe–Nphase(s) in principle can decompose underneath thesurface causing N2 gas development and thus the for-mation of pores. Ideally the kinetics of this processshould be infinitely slow at the nitriding temperature.

Now, a closer look at the establishment of equilibrium (2) isnecessary. At the surface of the specimen ammonia mole-cules are adsorbed and dissociate by stepwise removal ofhydrogen atoms [9]. This leads to adsorbed, individual ni-trogen atoms at the surface, Nads:

NH3 ! Nads þ 3=2H2 ð14Þ

Next, two routes can be followed by the adsorbed nitrogenatoms: they can dissolve into the solid substrate, which isthe desired effect:

Nads ! ½N ð15Þ

or they recombine at the surface and desorb, which counter-acts nitriding:

Nads þ Nads ! N2 " ð16Þ

Establishment of equilibrium (2) requires the establishmentof equilibrium for both reactions (14) and (15), the sum ofwhich yields the net reaction (2). In that case, the forwardand backward reactions according to Eq. (14) and accord-ing to Eq. (15) are equal. Evidently, this can only be rea-lized if recombination and desorption of Nads according toEq. (16) occurs with a negligibly slow rate as compared toEq. (14) and Eq. (15), but also requires that the diffusionof [N] dissolved in Fe adjacent to the surface to largerdepths is negligible (which holds if the specimen has beenhomogeneously nitrided and contains the maximum possi-ble amount of nitrogen corresponding with the prevailingnitriding conditions). Only under these conditions is thechemical potential (the nitriding potential) of nitrogen inthe gas atmosphere, as determined from the NH3 and H2contents in the gas atmosphere, equal to the chemical po-tential of nitrogen in the solid substrate at its surface, i. e. lo-cal equilibrium at the gas–solid interface has been estab-lished (see Fig. 1).



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However, if the recombination and desorption of the ad-sorbed nitrogen according to Eq. (16) occurs with a non-negligible rate,4 as compared to the rate of nitrogen uptakeaccording to Eq. (14) (assuming the equilibrium (15) is al-ways established), then the chemical potential of the dis-solved nitrogen at the surface of the specimen is lower thanthat determined from the NH3 and H2 contents in the gas at-mosphere (cf. Eq. (2)).In this case, if no significant diffusion of dissolved nitro-

gen into the bulk of the specimen occurs (see above), atime-independent situation (at constant temperature andconstant pressure) at the surface is established, as a conse-quence of equal rates of nitrogen uptake (Eq. (14); equilib-rium (15) is always established) and nitrogen desorption(Eq. (16)). Thus, although no equilibrium of the surface ofthe solid with the gas atmosphere according to only Eq. (2)is established, a so-called stationary state at the gas–solidinterface has been realized characterized by a content ofdissolved nitrogen in the solid substrate at the surface smal-ler than would occur if equilibrium at the gas–solid inter-face would prevail according to only Eq. (2). This state ofaffairs is illustrated in Fig. 1 [6, 9] and treated in more de-tail in the Appendix (see below).The tendency for recombination and desorption of Nads

becomes significant above about 580 8C for pure iron assubstrate. The difference between the equilibrium concen-tration, cN,eq, and the lower stationary value of the concen-tration, cN,st, at the surface of the nitrided component, in-creases with increasing temperature and nitriding potential(see Fig. 2; calculated following the procedure describedin the Appendix and adopting the reaction rate constantsgiven in Tables A1 and A2 of the Appendix). Then, the re-sult obtained for the content of dissolved nitrogen in the sol-id at the surface, i. e. cN,st, does not correspond with thechemical potential of nitrogen in the gas atmosphere, as de-termined from the NH3 and H2 contents in the gas atmo-sphere and as expressed by the nitriding potential (cf.Eqs. (4–6)). Clearly, in view of the results shown inFig. 2, at the usual nitriding temperatures (500 8C–580 8C)

local equilibrium at the surface of at least pure iron can beanticipated, provided the diffusion rate of the absorbed ni-trogen to larger depths is negligible as compared to the ni-trogen-uptake rate, i. e. only at later stages of nitriding. Athigher temperatures and/or at higher concentrations of dis-solved nitrogen, corresponding with relatively high nitrid-ing potentials, the contribution of the recombination anddesorption according to Eq. (16) can no longer be ne-glected: at a nitriding temperature of, for example, 580 8C(a nitriding temperature applied in practice) the nitridingpotential should not be larger than 6.1 atm–1/2 (correspond-ing to a, non-exceptional, 75% NH3 – 25% H2 gas mixture)in order that the difference of cN,eq and cN,st is smaller than1% relatively. Under the latter conditions nitrogen ferriteis likely not the \equilibrium" phase at the surface of thespecimen/component. Under conditions such that, in parti-cular, e iron nitride occurs at the surface of the specimen/component, a stationary state at the surface may already oc-cur at temperatures distinctly lower than 580 8C (see the ex-perimental results presented in Fig. 3 of Ref. [6]).The above consideration has as consequence that the

phase boundaries presented in the \potential", Lehrer dia-gram (cf. Figure 3) above about, 580 8C (and above lowertemperatures at high nitriding potentials as corresponding



Fig. 1. Concentration of dissolved N (at surface), cN, plotted againstnitrogen uptake rate, dcN/dt > 0, for nitriding in an NH3–H2 gas mix-ture according to Eq. (2) and nitrogen desorption rate, dcN/dt < 0, ac-cording to Eq. (16). The stationary state is given by dcN/dt|Eq. (2) +dcN/dt|Eq. (16) = 0 (see also below Eq. (A7)) [6].

(a)

(b)

Fig. 2. (a) Relative difference in the equilibrium concentration, cN,eq,and the stationary state value, cN,st, depending on the nitriding tempera-ture. (b) Enlargement of part of (a).

4 The nitriding reaction according to Eq. (16) can be neglected due toits comparatively low rate [10] and the flowing gas atmosphere main-taining pN2 0.

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to the development of iron nitrides) are incorrect, as the ex-perimentally determined nitriding potential for the phaseboundary (at a certain temperature) corresponds to a sta-tionary state rather than a (local) equilibrium. The true ni-triding potential, characterizing the chemical potential ofthe hypothetical gas atmosphere in equilibrium with the sol-id (at the phase boundary) and as given by the NH3 and H2contents of that hypothetical gas atmosphere, will be smal-ler than the experimentally determined one: see the resultsshown for the a/c phase boundary in Fig. 3a; the effectmay be considerably larger for the c’/e phase boundary, asindicated in the previous paragraph.Corrections for the occurrence of stationary states instead

of (local) equilibria will be applied to the data obtained inthis work for the solubility data of nitrogen in ferrite andthe phase-boundary data in the Lehrer diagram.However, the \normal", temperature, T, versus composi-

tion, cN, Fe–N phase diagram is not affected by the occur-

rence of a stationary state at the surface of the specimen:the (T, cN) data points are determined from the (T, rN) datapoints for the phase boundary concerned in the Lehrer dia-gram using the determined absorption function (see Sec-tion 5.3). Following this procedure, if the uncorrected (inthe above sense) Lehrer diagram is used, the uncorrected ab-sorption function should also be utilized, whereas if the true(i. e. corresponding to genuine (local) equilibrium) Lehrerdiagram is used, the absorption function corresponding with(local) equilibrium should be utilized (see Section 4.2).The quantitative results presented above pertain to ni-

trided pure ferrite (Fig. 2). The necessary kinetic data forsuch calculations, for the rates of reactions (14), (15) and(16), are available for ferrite and austenite substrates [9–12] (but see the Appendix). Such kinetic data are not avail-able for the case that iron nitrides have formed. Therefore,as compared to nitrogen ferrite, the occurrence of the re-combination and desorption reaction (Eq. (16)) and the oc-



(a) (b)

(c)

Fig. 3. Re-evaluation of the a/c’ and a/c phase boundaries in the (T,rN), Lehrer diagram. (a) The a/c and a/c’ boundaries as determined inthis work: the experimental data (above, approximately 580 8C) repre-sent stationary states; the calculated (hypothetical) equilibrium states(see Section 3 and the Appendix) are represented by the correspondinglines in the figure. (b) Presentation of all experiments performed to de-termine the a/c’ and a/c phase boundaries. (c) Enlargement of (b)around the a/c/c’ triple point.

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currence of a stationary state may become significant abovedifferent temperature and nitriding potential values.

4. Experimental

4.1. Specimen preparation and gaseous nitriding

Iron pellets (Alpha Aesar, 99.98 wt.% Fe – for chemicalanalysis, see Table 1) were melted in an induction furnaceunder argon atmosphere. The rods obtained after castingand solidification were cold-rolled to sheets of a thicknessof 300 mm. Specimens (15 · 20 mm) were cut out from therolled sheets, ground and polished (final stage 1 mm dia-mond paste), with ultrasonic cleaning in ethanol after eachstage. Next the specimens were recrystallized in a pure hy-drogen flow (200 ml min–1) at 700 8C for 1 h.For gaseous nitriding, a specimen was suspended in the

middle of a vertical quartz-tube furnace by a quartz fiber.To achieve a high precision in the control of the process

temperatures, the thermocouples used were calibrated uti-lizing a series of well-defined melting points of pure ele-ments and alloys; the error of the process temperature issmaller than ±1 8C. To assure a constant temperature at thespecimen position in the furnace, temperature profilesalong the furnace axis were measured. It could be shownthat the temperature was homogeneous within the (length)range of the specimen.The gas flows were adjusted by mass flow controllers

which exhibit an accuracy of 0.1% of the selected flow-ratevalue. As a nitrogen providing medium, an ammonia–hydro-gen (both 99.999 vol.%) gas mixture, with the required ratio(to maintain a specified nitriding potential) was used. Theoverall gas flow amounted to 500 ml min–1, which equatesto a linear flow rate of 13.5 mm s–1 at room temperature forthe applied tube furnace with diameter of 28 mm [13]; there-by any thermal decomposition of NH3 in the furnace is negli-gible (see discussion in Section 3 and Ref. [6]).After the nitriding treatments, by shearing of the quartz

fiber, the specimen dropped down into a bottle containingwater at room temperature and was thus quenched in orderto retain the state produced at the nitriding temperature atthe end of the nitriding time. The nitriding time was chosensuch (usually 24 h at T > 450 8C) that the specimen washomogeneously nitrided. In fact a shorter time would havebeen sufficient, as follows from a calculation using nitrogendiffusion-coefficient data from Ref. [14]. Indeed, prolonga-tion of the nitriding time to 48 h yielded the same amount ofnitrogen taken up as measured after 24 h. Only forT £ 450 8C longer nitriding times were applied (64 h at400 8C) which assured homogenous nitriding.

4.2. Analyzing/characterization methods

The mass of the specimens was measured before and afterthe nitriding treatments using a \Mettler Toledo microba-

lance UMX2". Before each measurement the specimenwas dried for a couple of hours in a drying cabinet at about50 8C. The mass change (upon nitriding) was determinedas the calculated average value of about 8 measurementsfor each specimen. The mass gain after nitriding was as-cribed to the nitrogen taken up by the specimen. X-ray dif-fraction (XRD) measurements for phase analysis were per-formed before and after the nitriding treatment, using a\Phillips X’Pert Multi-Purpose Diffractometer" using Co-Ka radiation and applying Bragg–Brentano geometry.Light-optical microscopy as well as Vickers hardness

measurements were performed on cross-sections of the spe-cimens, using a \Zeiss Axiophot microscope" and a \LeicaVMHT MOT hardness tester", respectively. These cross-sections were prepared by cutting a piece of the specimens,after nitriding and embedding in \Polyfast" (BuehlerGmbH). The embedded specimen was ground, polishedand cleaned ultrasonically as described in Section 2.1. Thegrain structure was made visible in the cross-section byetching in a 2% Nital (2 vol.% HNO3 in ethanol) solutionfor 1 minute.

5. Results and discussion

In this work the a/c and a/c’ phase boundaries of the Lehrerdiagram were determined by the following approach:(i) a certain nitriding potential was selected;(ii) nitriding experiments with this nitriding potential were

performed at different nitriding temperatures;(iii) the phase(s) present after each nitriding experiment

was (were) determined by XRD;(iv) the data points, i. e. the (T, rN) points in the Lehrer dia-

gram, were plotted in the Lehrer diagram with indica-tion of identified phase(s).

(v) the phase-boundary (T, rN) was taken to run betweenneighboring data points at the same rN pertaining todifferent phase constitutions (Section 5.2).

Using all solubility measurements, an absorption functionfor the aFe[N] phase was determined in the temperaturerange 400 8C–680 8C (see Section 5.2). By insertion of thenitriding potential at the phase-boundary temperature, de-termined as described above, into the absorption function,an (analytical) description of the corresponding phaseboundary in the (T, cN) diagram was obtained (Section 5.3).Following the approach indicated above, phase bound-

aries were experimentally determined by nitrogen uptakeand phase-constitution measurements in the temperaturerange of 400 8C–680 8C. Thus, 12 phase-boundary loca-tions were determined in the temperature range of 500–650 8C: five pertaining to the equilibrium of aFe[N] andcFe[N]; seven pertaining to the equilibrium of aFe[N] andc’Fe4N1–x. The nitriding potentials, temperatures and nitro-gen contents of ferrite at these equilibria, as well as theiruncertainties, derived from the separation of the adjacent



Table 1. Chemical analysis of the pure iron used in this work. Metal contents were determined from pellets, whereas non-metal contentswere determined from the already cold-rolled material after melting, casting and solidification, but before the recrystallization treatmentwas performed (see Section 4.1).

element N C O S Mn Ni Co Cr V Fe

content (wt.%) 0.0016 0.0015 0.022 0.003 0.0015 0.0015 0.001 0.0003 0.0003 Bal.

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data points at opposite sides of the phase boundary, aregathered in Tables 2 and 3.The apparent nitriding potentials, as determined experi-

mentally, for the a/c’ and a/c phase boundaries in theLehrer diagram were corrected for the genuinely occurringstationary states, instead of (local, at the surfaces of the spe-cimens) equilibria. Such a correction is straightforwardlyperformed by insertion of the experimentally determinedvalue of the nitrogen solubility in ferrite at the phase bound-ary into the corrected absorption function (17c) (see Sec-tion 5.3). The thus corrected nitriding potentials, albeit hy-pothetical, representing genuine (local) equilibrium, forthe a/c’ and a/c phase boundaries, are also presented in Ta-bles 2 and 3.

5.1. Lehrer diagram

An overview of the new a/c and a/c’ phase-boundary data(both for the experimentally occurring stationary statesand for the (hypothetical) equilibria; see above and Sec-tion 3), in comparison with those from Refs. [4] and [15],is provided by Fig. 3a for temperatures above 500 8C5. Thedata points of the a/c and a/c’ phase boundaries presentedin this paper were derived from 58 experiments, with resultspresented in Fig. 3b.The triple point in the Lehrer diagram, where the a, c’

and c phase fields meet, i. e. where the a/(a + c’) and a/(a + c) phase boundaries meet in the Fe–N phase diagram(defining the composition of a as part of the eutectoid reac-tion c ? a + c’), is determined by the intersection of the a/c and a/c’ phase boundaries in the Lehrer diagram. By gra-phical extrapolation of the a/c and a/c’ phase boundaries,as determined in this work, it thus follows that the triplepoint is given by T = 593.0 8C and rN = 0.139 atm–1/2 (sta-tionary state), and T = 593.0 8C and rN = 0.135 atm

–1/2

(equilibrium) (cf. Fig. 3c). This can be compared with the

results of Refs. [4] and [15]. The here determined coordi-nates for the triple point differ distinctly from those ofRef. [15] and are close to those of Ref. [4]. The here deter-mined a/c phase boundary differs from those presented inRefs. [4] (experimental data) and [15] (calculated (CAL-PHAD-approach) assessment); the here determined a/c’phase boundary is close to that of Ref. [4].

5.2. Absorption function for a-Fe[N]

The results of all nitrogen-solubility measurements of thea-Fe[N]-phase for T > 400 8C are gathered in Fig. 4a wherethe data are presented in a plot of ln(cN,a/rN) vs. 1/T (cf.Eq. (9)). It follows that the nitrogen solubility of the a-Fe[N] phase cannot be described by Eq. (9) over the entire,here investigated large temperature range (400–680 8C).Absorption functions on the basis of Eq. (9) have been de-termined before [9, 17–19], which results were evaluatedin Refs. [20] (300 8C < T < 590 8C) and [21](300 8C < T < 600 8C). The differences between these eva-luations are small.The present data can be represented by straight lines in

two temperature ranges (least squares analysis; see Fig. 4a):

T < 500 C: ln cN1rN¼ 17:48 9267:7

Tð17aÞ

500 C < T < 680 C: ln cN1rN¼ 18:7 10164

T

ð17bÞ

where cN, rN and T are given in units of at.% N, Pa–1/2 andK, respectively.The result for T < 500 8C agrees well with those pre-

sented in Refs. [20] and [21] (see above).The absorption data presented in Fig. 4a in fact represent

stationary states, as discussed in Section 3. In particularabove 600 8C a correction to obtain absorption data repre-



Table 2. The equilibrium of a-Fe[N] and c-Fe[N] (and the corresponding gas atmosphere).

nitriding potential(equilibrium, calc.)

(atm–1/2) 0.049 0.066 0.100 0.113 0.133

nitriding potential(stationary state, exp.)

(atm–1/2) 0.052 0.070 0.104 0.117 0.137

temperature (8C) 654.5 634.5 611.5 604.5 594.5error ( ± ) (8C) 1.5 1.5 1.5 1.5 1.5

nitrogen content (at.%) 0.360 0.386 0.423 0.434 0.445error ( ± ) (at.%) 0.0064 0.0070 0.0082 0.0086 0.0090

Table 3. The equilibrium of a-Fe[N] and c’-Fe4N1–x (and the corresponding gas atmosphere).

nitriding potential(equilibrium, calc.)

(atm-1/2) 0.136 0.138 0.144 0.157 0.180 0.201 0.253

nitriding potential(stationary state, exp.)

(atm-1/2) 0.140 0.142 0.148 0.160 0.182 0.201 0.250

temperature (8C) 591.5 588.0 584.5 574.5 558.0 544.5 521.5error ( ± ) (8C) 1.5 2.0 1.5 1.5 2.0 1.5 1.5

nitrogen content (at.%) 0.436 0.422 0.419 0.394 0.353 0.319 0.277error ( ± ) (at.%) 0.0089 0.0116 0.0087 0.0084 0.0104 0.0073 0.0067

5 Some data at temperatures below 400 8C were presented inRef. [16].

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senting (local, at the surface) equilibrium is necessary.Using the rate data for the reactions represented inEqs. (14–16) (see Appendix), such a correction can bestraightforwardly performed: cN,eq follows from cN,st usingEq. (A8). The result is shown in Fig. 4b. This of courseleads to (even) a little more curvature in the absorptionfunction. As a result it is now obtained for the higher tem-perature range:

500 C < T < 680 C: ln cN1rN¼ 19:16 10579

T

ð17cÞ

where cN, rN and T should be given in at.% N, Pa–1/2 and K,respectively. The upward curvature, from the straight linedescribed by Eq. (17a) in Fig. 4 can be considered as a con-sequence of a dominating (weak) dependence on tempera-ture of the enthalpy of nitrogen absorption (cf. Eq. (9)).

5.3. Fe–N phase diagram

The determination of phase-boundary points (T, cN) for thea/(a + c) and a/(a + c’) phase boundaries follows by substi-tution of the corresponding (T, rN) data for the a/c and a/c’phase boundaries in the Lehrer diagram (see discussion inSection 3) into the description for the absorption function(Eq. (17)). The resulting (T, cN) data points, including theend-point of the eutectoid line, where the a/(a + c) and a/(a + c’) phase boundaries meet in the Fe–N phase diagram

(determined as 593.0 8C at cN = 0.441 at.% from the eutec-toid (T, rN) data obtained as described in Section 5.1), areshown in Fig. 5. The experimental data from previousworks, as well as the calculated phase boundaries accordingto evaluations presented in Refs. [20] and [22] are also indi-cated in Fig. 5 [23–32].The here determined coordinates of the end-point of the

eutectoid line, where the a/(a + c) and a/(a + c’) phaseboundaries meet, differ distinctly from those according tothe evaluations of Refs. [20] and [22]. Also the a/(a + c’)and a/(a + c) phase boundaries as determined here differsignificantly from those presented in Refs. [20] and [22].In particular the a/(a + c’) phase boundary lies at higher ni-trogen contents than as suggested by the evaluations ofRefs. [20] and [22].

5.4. Heat of solution/reaction

From the slope of the straight lines in Fig. 4 (for ln(cN,a/rN)vs. 1/T) the enthalpy of dissolution of nitrogen in a-Fe canbe determined (Eq. (9)). Thus, below and above 500 8C,DH0N!a values of 77.1 kJ mol

–1 and 88.0 kJ mol–1 are ob-tained from Eqs. (17a) and (17c), respectively. The resultfor T < 500 8C well agrees with those of the evaluations inRefs. [20] and [21] (cf. Section 5.2).The enthalpy for the reaction a? c’ can be determined

from the plot of lncN,a/c’ vs. 1/T (Eq. (12)). Such a plot isshown for the a/(a + c’) data points obtained in this work inFig. 6a [14, 21, 23–27, 29–32]. Indeed, a straight line is ob-tained in this plot. From the slope of this straight line DH0a=c0



(a) (b)

Fig. 4. (a) The nitrogen-absorption function for a-Fe[N] in the temperature range 400–680 8C, as presented in a plot of ln(cN,a/rN) vs. 1/T (cf. Eq.(9)). The straight lines shown correspond with least squares fittings to the experimental data for the temperature ranges indicated. (b) Nitrogen-ab-sorption function for a-Fe[N] in the temperature range 550–680 8C. The experimental values of the nitriding potential, rN, above a temperature ofapproximately 580 8C represent stationary states. The corresponding values of rN representing the (hypothetical) equilibria were calculated follow-ing the procedure described in the Appendix (see Section 3).

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can be determined as –39.2 kJ mol–1 (525–585 8C; leastsquares analysis). The phase-boundary data points of otherworks are gathered in Fig. 6b, where two dashed lines havebeen drawn to enclose most of these data. According to thesetwo straight lines, DH0a=c0 values according to these previousliterature data vary between –23.0 and –43.0 kJ mol–1.

6. Conclusions

The evaluation of the experimental data of this work leadsto the following results:(i) Parts of the phase boundaries in the experimentally

determined Lehrer diagram represent stationary statesrather than equilibria.

(ii) Procedures have been devised to calculate the (hy-pothetical) equilibrium values of the concentration ofdissolved nitrogen and the corresponding nitridingpotential from the experimentally determined valuespertaining to a stationary state.

(iii) The solubility of nitrogen in ferrite (a-Fe) is deter-mined by:

T < 500 C:

ln cN1rN¼ 17:48 9267:7

T

500 C < T < 680 C:

ln cN1rN¼ 18:7 10164

Tðstationary stateÞ

500 C < T < 680 C:

ln cN1rN¼ 19:16 10579

Tðlocal equilibriumÞ

with cN, rN and T expressed in at.% , Pa–1/2 and K, re-spectively.

(iv) The triple point where the a, c and c’ phase fieldsmeet in the (T, rN), Lehrer diagram is given byT = 593.0 8C and rN = 0.139 atm–1/2 (stationary state),and T = 593.0 8C and rN = 0.135 atm–1/2 ((local) equi-librium). This corresponds with the end-point of theeutectoid line in the \normal" (T, cN), phase diagram,where the a/(a + c’) and a/(a + c) phase boundariesmeet, and which is given by T = 593.0 8C and cN =0.441 at.%.

(v) The courses of the a/c and a/c’ phase boundaries inthe (T, rN), Lehrer diagram, close to the triple (a/c/c’) point, deviate significantly from the old, experi-mental data presented in Ref. [4] and the newer, calcu-lated data presented in Ref. [15].

(vi) The courses of the a/(a + c’) and a/(a + c) phaseboundaries in the (T, cN), Fe–N phase diagram, closeto the end-point of the eutectoid line, where the a/(a + c’) and a/(a + c) phase boundaries meet, deviatesignificantly from earlier evaluations such as thosepresented in Refs. [18] and [21].

(vii) The enthalpy of dissolution of N in a-Fe dependssomewhat on temperature and has been determinedas 77.1 kJmol–1 for T < 500 8C and as 88.0 kJ mol–1

for 500 8C < T < 680 8C.



Fig. 5. Re-evaluation of the a/(a + c’) and a/(a + c) phase boundaries in the (T, cN), Fe–N phase diagram. The bold full lines represent the resultsof this work, as derived from the nitrogen-absorption function determined for a-Fe[N] (Fig. 4) after substitution of the rN values at the phase bound-aries concerned, as determined for the (T, rN), Lehrer diagram (Fig. 3a); see text.

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(vii) The enthalpy for the reaction at the a/c’ phase bound-ary has been determined as –39.2 kJ mol–1 (525 8C–585 8C).

The authors thank Dipl.-Ing. P. Kress (Max Planck Institute for Intelli-gent Systems) for technical assistance.



(a)

(b)

Fig. 6. Determination of the enthalpy for thereaction a?c’ by plotting lncN,a/c’ vs 1/T (cf.Eq. (13)). (a) The reaction enthalpy can be de-termined from the slope of the straight lineplotted in (a) (least squares fitting to the ex-perimental data obtained in this work). (b)Corresponding literature data, also at lowertemperatures (see text).

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References

[1] E.J. Mittemeijer: Fundamentals of Materials Science, Springer-Verlag, Berlin-Heidelberg (2010). PMid:21328479

[2] S. Lampman: Introduction to Surface Hardening of Steel, ASMHandbook Vol 4, Heat Treating, Metals Park, Ohio, ASM Interna-tional 4 (1997) 259.

[3] T.B. Massalski, H. Okamoto: Binary Alloy Phase Diagrams, ASMInternational, USA (1990).

[4] Lehrer, E: Z. Elektrochem. Angew. P. 36 (1930) 383.[5] E.J. Mittemeijer, J.T. Slycke: Surf. Eng. 12 (1996) 152.[6] E.J. Mittemeijer, M.A.J. Somers: Surf. Eng. 13 (1997) 483; see

also: M.A.J. Somers, B.J. Kooi, L. Maldzinski, E.J. Mittemeijer,A.A. van der Horst, A.M. van der Kraan, N.M. van der Pers: ActaMater. 45 (1997) 2013.

[7] D.R. Gaskell: Introduction to the Thermodynamics of Materials,Taylor & Francis Inc. (2008).

[8] M. Fonovic, A. Leineweber, E.J. Mittemerijer: submitted for pub-lication.

[9] H.J. Grabke: Berich. Bunsen Gesell., 72 (1968) 533.[10] H.J. Grabke: Berich. Bunsen Gesell., 72 (1968) 541.[11] H.J. Grabke: Arch. Eisenhuettenwes., 46 (1975) 75.[12] H.J. Grabke: Z. Phys. Chem. Neue Fol., 100 (1976) 185.

DOI:10.1524/zpch.1976.100.3-6.185[13] M. Nikolussi, A. Leineweber, E.J. Mittemeijer: Acta Mater. 56

(2008) 5837. DOI:10.1016/j.actamat.2008.08.001[14] J.D. Fast, M.B. Verrijp: J. Iron Steel Inst., 176 (1954) 24.[15] A.F. Guillermet, H. Du: Z. Metallkd., 85 (1994) 154.[16] E.H. Du Marchie van Voorthuysen, N.C. Chechenin, D.O. Boer-

ma: Metall. Mater. Trans. A 33 (2002) 2593–2598.DOI:10.1007/s11661-002-0380-2

[17] H.H. Podgurski, H.E. Knechtel: Trans. Am. Inst. Min. Met. Eng.,245 (1969)1595.

[18] Z. Przylecki, L. Maldzinski: in Proc. 4th Int. Conf.: PolitechnikaPoznanska. (1987) 153.

[19] H.A.Wriedt, L.S. Darken, R.P. Smith: unpublished results (1964),as cited in Ref. [21].

[20] B.J. Kooi, M.A.J. Somers, E.J. Mittemeijer: Metall. Mater. Trans.A 27 (1996) 1063. DOI:10.1007/BF02649774

[21] H.A. Wriedt, N.A. Gokcen, R.H. Nafzinger: Bulletin of AlloyPhase Diagrams, 8 (1987) 355. DOI:10.1007/BF02869273

[22] J. Kunze: Nitrogen and Carbon in Iron and Steel Thermody-namics, Physical Research Vol. 16, Akademie-Verlag, Berlin(1990). PMid:2316057

[23] V.G. Paranjpe, M. Cohen, M.B. Bever, C.F. Floe: Trans. Am. Inst.Min. Met. Eng., 188 (1950) 261.

[24] N.S. Corney, E.T. Turkdogan: J. Iron Steel Inst., 180 (1955) 344.[25] L.J. Dijkstra: Trans. Am. Inst. Min. Met. Eng., 185 (1949) 252.[26] M. Nacken, J. Rahmann: Arch. Eisenhuettenwes., 33 (1962) 131.[27] K. Abiko, Y. Imai: Trans. Jpn. Inst. Met, 18 (1977) 113.[28] W. Pitsch, E. Houdremont: Arch. Eisenhuettenwes. 27 (1956) 281.[29] A. Burdese: La Metallurgia Italiana, 8 (1955) 357.[30] R. Rawlings, D. Tambini: J. Iron Steel Inst., 184 (1956) 302.[31] H.U. Åström, G. Borelius: Acta Metall., 2 (1954) 547.

DOI:10.1016/0001-6160(54)90076-0[32] G. Collette, C. Roederer, C. Crussard: Mem. Sci. Rev. Met., 58

(1961) 61.[33] Y. Imai, T. Masumoto, M. Sakamoto: Sci. Rep. Res. Tohoku A,

20 (1968) 1.[34] H.C.F. Rozendaal, E.J. Mittemeijer, P.F. Colijn, P.J. Van-

derschaaf: Metall. Trans. A 14 (1983) 395.DOI:10.1007/BF02644217

[35] P.B. Friehling, F.W. Poulsen, M.A.J. Somers: Z. Metallkd., 92(2001) 589.

(Received March 25, 2013; accepted June 17, 2013; onlinesince September 10, 2013)

Bibliography

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Dr. Ralf Erich SchacherlInstitute for Materials ScienceUniversity of StuttgartHeisenbergstr. 370569 StuttgartGermanyTel.: + 49711689-3314Fax: + 49711689-3312E-mail: [email protected]

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Appendix

Nitrogen-flux equations; derivation of the relation be-tween the equilibrium and stationary values of the nitro-gen concentrationThe concentration of nitrogen adsorbed at the Fe surface

is determined by the three reactions (14), (15) and (16):the dissociation/formation of NH3, the dissolution of ad-sorbed nitrogen in Fe and its reversal, and the dissociation/formation of N2. Since the dissolution and reformation ofadsorbed N is supposed to be very rapid, equilibrium (15)is assumed to be always realized.The net flux of nitrogen into the specimen/foil according

to only Eq. (14) (assuming equilibrium (15) to be estab-lished) then obeys [9]

Jð14Þ¼k0ð14Þ;

pNH3

pH2ð Þ32kð14Þ; cN pH2ð Þ2 ðA1Þ

with cN as the concentration of [N] in Fe adjacent to the sur-face and k

0ð14Þ; and kð14Þ; as the rate constants for ammonia

dissociation and ammonia formation, respectively, and¼ 2 or 3, depending on which hydrogenation step of ad-

sorbed N is rate-limiting at given pH2 and T.The net flux of nitrogen into the specimen/foil according

to only Eq. (16) (again assuming equilibrium (15) to be es-tablished) obeys [10]

Jð16Þ ¼ k0ð16Þ

11þ KcN

pN2 kð16ÞK1

1þ KcNc2N ðA2Þ

with the rate constants k0ð16Þ and kð16Þ for nitrogen dissocia-

tion and association, respectively, and the constant K de-scribing the segregation equilibrium (Eq. (15)) of dissolvedand adsorbed nitrogen [12]. Eq. (A2) holds for high cN asachieved upon nitriding with NH3/H2.If the formation of N2 can be neglected (for T < 580 8C

and rN < 6.1 atm–1/2, see Section 3) the total flux of nitrogen

into or out of the specimen is given by Eq. (A1) only. Then,equilibrium (14) can be established at constant pNH3 and pH2(as realized by a flowing gas atmosphere, cf. Section 3), im-



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plying that the specimen/foil is homogeneously nitridedand Jð14Þ ¼ 0. From Eq. (A1) it then follows that

cN ¼ cN;eq ¼k0ð14Þ;kð14Þ;

pNH3p3=2H2

ðA3Þ

for the equilibrium concentration of [N]. Eq. (A3) isequivalent to Eq. (7): both are valid only if N2-formationis negligible. Substitution of k

0ð14Þ; pNH3 by cN;eqkð14Þ; p

3=2H2

in Eq. (A1) yields the alternative flux equation (indicatedby the superscript \*")

Jð14Þ ¼ kð14Þ; pH2ð Þ2 cN;eq cN ðA4Þ

Thus the ammonia-related nitrogen flux can be expressedusing either the rate constants k

0ð14Þ; and kð14Þ; (Eq. (A1)),

or the rate constant kð14Þ; for ammonia formation and theequilibrium concentration cN,eq (Eq. (A4)).If formation of N2 cannot be neglected, cN is also influ-

enced by reaction (16). This effect can be accounted for intwo different ways.Firstly, the net nitrogen flux can be obtained by adding

J(16), the flux of nitrogen into the specimen/foil accordingto only Eq. (16) (assuming equilibrium (15) to be estab-lished), to the expression for J(14) given by Eq. (A1). Setting

pN2 ¼ 0, as holds for typical nitriding conditions, the result-ing net flux equation becomes

Jð14Þþð16Þ¼ k0ð14Þ;

pNH3

pH2ð Þ32kð14Þ; cN pH2ð Þ2

kð16ÞK1

1þ KcNc2N ðA5Þ

Time independence is now reached once Jð14Þþð16Þ ¼ 0,which happens at cN ¼ cN;st < cN;eq: a stationary state dif-ferent from establishment of only the equilibrium (14).Note that according to Eq. (A5), cN;st is no longer propor-

tional to rN ¼pNH3p3=2H2

, i. e. in principle the experimental data

cannot be described by Eq. (A3), Eq. (7) or Eq. (9). In ana-logy to the procedure to arrive at Eq. (A4), equatingJð14Þþð16Þ with zero for cN ¼ cN;st leads to a relation fork0ð14Þ; pNH3 which can be substituted into Eq. (A5):

Jð14Þþð16Þ ¼ kð14Þ; pH2ð Þ2 cN;st cN þ cNÞ

þ kð16ÞKc2N;st

1þ KcN;stc2N

1þ KcN

!ðA6Þ



Table A1.

Ammonia dissociation Ammonia formation

k0ð14Þ; ¼2

(mol s–1 cm–2 atm–1/2k0ð14Þ; ¼3

mol s 1 cm 2 atm 1ð Þkð14Þ; ¼2

mol s 1 cm 2ð Þkð14Þ; ¼3

mol s 1 cm 2 atm 1ð Þ

9:8 exp138:5 kJmol 1

RT1:1 exp

120:3 kJmol 1

RT9 10 2 exp

64:2 kJmol 1

RT1 10 2 exp

46:0 kJmol 1

RT

Table A1: Rate constants for reaction (14) (ammonia dissociation and formation) according to Ref. [9] (cf. comment in caption of Ta-ble A2). The rate constants k

0ð14Þ; and kð14Þ; and cN,eq are related (see Eq. (A3)). In Ref. [9] experimental values for kð14Þ; and cN,eqwere

determined and using Eq. (A3) a value for k0ð14Þ; was derived. The current experiments provided a set of cN,eq values different from those

in Ref. [9]. This would mean that a different value for k0ð14Þ; would follow from the current cN,eq values and the value of kð14Þ; as given

in Ref. [9]. It is noted that this ambiguity in k0ð14Þ; does not affect all the calculations in this paper as calculations are performed indepen-

dent of k0ð14Þ; (cf. Eq. (A8)).

Table A2.

Nitrogen dissociation Nitrogen formation

k0ð16Þ;a-Fe

mol s 1 cm 2 atm 1ð Þk0ð16Þ; -Fe

mol s 1 cm 2 atm 1ð Þkð16Þ;a-Fe

cm4 mol 1 s 1

kð16Þ; -Fecm4 mol 1 s 1

44 exp235:1 kJmol 1

RT12 exp

219:7 kJmol 1

RT8:2 107 exp

171:5 kJmol 1

RT3:3 108 exp

232:2 kJmol 1

RT

Table A2: Rate constants for reaction (16) (nitrogen dissociation and formation) according to Ref. [10]. The rate constants for nitrogen

formation were derived in the present work from the relation k ¼ v2

k0pN2;refwhere v is the activity coefficient for nitrogen in ferrite and with

pN2;ref ¼ 1 atm (cf. Eq. (16) in Ref. [10]) using the experimental data reported in Ref. [10]. The values of k reported in Ref. [11] on the ba-sis of experimental data in Ref. [10] were found to be erroneous (in particular the pre-exponential factors; the activation energies reportedin Ref. [11] agree well with the values derived here from the data in Ref. [10]). The constant K (cf. Eq. (A2)) in wt.%–1was adopted as gi-ven in Ref. [12] by the equation log10 K=wt:%

1ð Þ ¼ 5 730 K=T 3:98 with T in K. Note that for the backward reactions of Eqs. (14)and (16), using the rate constants kð14Þ; ¼2 or kð14Þ; ¼3 as given in Ref. [9] (Table A1) and kð16Þ;a-Fe as derived in this work from the datain Ref. [10] (see above; numerical data below), the calculated fluxes were about twenty times higher than the experimental data depictedin Fig. 5 of Ref. [9]. This is assumed to be the result of an erroneous scaling factor in Fig. 5 of Ref. [9].

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Thus Jð14Þþð16Þ can be expressed either by the three rate con-stants k

0ð14Þ; ; kð14Þ; and kð16Þ (Eq. (A5)) or by the two rate

constants for denitriding, kð14Þ; and kð16Þ, and the saturationconcentration pertaining to the stationary state cN,st(Eq. (A6)).Secondly, to obtain the net nitrogen flux into the speci-

men/foil, the correction term J(16) can be added toEq. (A4). Setting again pN2 ¼ 0, the resulting alternativeflux equation (indicated by the superscript \*") now be-comes

Jð14Þþð16Þ ¼ kð14Þ; pH2ð Þ2 cN;eq cN kð16ÞK1

1þ KcNc2N

ðA7Þ

Time independence is reached once Jð14Þþð16Þ ¼ 0 withcN ¼ cN;st. With Jð14Þþð16Þ ¼ 0, a relation is obtained fromEq. (A7) that relates the stationary, experimentally accessi-ble, concentration cN,st and the (hypothetical) equilibriumconcentration cN,eq:

cN;eq ¼ cN;st þkð16ÞKc

2N;st

kð14Þ; pH2ð Þ2 1þ KcN;stðA8Þ

The results shown in Fig. 2 were calculated, on the basis ofEq. (A8), using the data in Tables A1 and A2.To correct the experimental absorption function for the

occurrence of stationary states, the value of cN,eq, pertainingto the nitriding potential and temperature for which the val-ue of cN,st was measured, has to be determined. This valueof cN,eq immediately follows by application of Eq. (A8).To correct the experimental Lehrer diagram for the oc-

currence of stationary states, it is recognized that the solidphases at the surface (in this study at least one of these isthe nitrogen-ferrite phase) are in equilibrium with eachother, but that a stationary state prevails at the interfacewith the surrounding gas atmosphere. The nitriding poten-tial representing equilibrium of the solid with the gas phase

at the surface is obtained by adopting the amount of dis-solved nitrogen in nitrogen ferrite at the solid state phaseboundary (i. e. cN,st as obtained by interpolation; cf. begin-ning of Section 5) as a value for cN,eq and inserting that val-ue in the corrected absorption function. On this basis, the\equilibrium" nitriding potentials in Tables 2 and 3 werecalculated from the experimental nitriding potentials repre-senting the stationary states.The net nitrogen fluxes introduced above and as pertain-

ing to the reactions Eq. (14) and (16) describe the changein [N]-concentration, cN, by nitrogen exchange betweenthe gaseous phase and Fe adjacent to the surface. Upon ni-triding (or de-nitriding) before reaching the saturation con-centration cN,eq or cN,st, the change in cN is also dependenton the diffusion of [N] into (or out of) the specimen bulk.The diffusion of [N] in the specimen, represented by theflux Jdiff, can be described by Fick’s 2nd law (in one dimen-sion)

qcNqt¼ q

qxDqcNqx

ðA9Þ

withD as the diffusion coefficient of [N] in Fe and x and t asdepth coordinate and time, respectively. The gas-relatedfluxes Jð14Þþð16Þ and the diffusion flux Jdiff are coupled bythe mass conservation condition

Jdiff jx¼0 ¼ DqcNqx x¼0

¼ Jð14Þþð16Þ ðA10Þ

which must be fulfilled at x = 0. As a result, depending onthe boundary conditions of the (de-)nitriding experiment,such as sample size and geometry and nitriding parameters(rN, T), the time variation (at constant temperature) of thenitrogen-concentration profile in the specimen can be cal-culated [34, 35], which is associated with a gradual increaseof the concentration of [N] in Fe adjacent to the surface un-til the foil has been homogeneously nitrided, which situa-tion represents either \equilibrium" or a \stationary state".



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solubility of nitrogen in ferrite; the fe–n phase diagram

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